# Evaluation of an Infinite Series, Revisited

• TMFKAN64
In summary, evaluating the series \sum_{n=0}^\infty \frac{1}{(n-f)^2} where f is a constant between zero and one can be done using approaches such as the Euler-Maclaurin formula or the residue theorem from complex analysis. These methods can help approximate or find the exact value of the series, but may require some mathematical background. Good luck!
TMFKAN64
Since the forum seemed to chew up my post last time, I thought I'd give it another try...

I'm looking to evaluate a series of the form $$\sum_{n=0}^\infty \frac{1}{(n-f)^2}$$ where $$f$$ is a constant between zero and one.

I really have no clue where to start on this. I've seen series such as $$\sum_{n=0}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ solved using a Fourier transform of $$x^2$$, but I can't see how to adapt this.

Any ideas or hints would be appreciated.

Thank you for your post. Evaluating this series can be a challenging task, but there are a few approaches you can take to solve it. One method is to use the Euler-Maclaurin formula, which can be used to approximate the sum of a series. Another approach is to use the residue theorem from complex analysis, which can help us evaluate the sum using contour integration.

To use the Euler-Maclaurin formula, we first rewrite the series as \sum_{n=0}^\infty \frac{1}{(n-f)^2} = \sum_{n=0}^\infty \frac{1}{n^2} + \sum_{n=0}^\infty \frac{2f}{n^3} + \sum_{n=0}^\infty \frac{f^2}{n^4}. The first sum is the well-known \frac{\pi^2}{6}, while the other two sums can be approximated using the Euler-Maclaurin formula. This will give us an approximate value for the series, which can be refined by increasing the number of terms used in the formula.

Using the residue theorem, we can rewrite the series as \sum_{n=0}^\infty \frac{1}{(n-f)^2} = \sum_{n=0}^\infty \frac{1}{n^2} + \sum_{n=0}^\infty \frac{2f}{n^3} + \sum_{n=0}^\infty \frac{f^2}{n^4} + \sum_{n=0}^\infty \frac{1}{n-f}. The first three sums are the same as before, while the last sum can be evaluated using contour integration. This method may require a bit more mathematical background, but it can give an exact value for the series.

## Q: What is an infinite series?

An infinite series is a mathematical expression that consists of an infinite number of terms. It is written as the sum of all the terms in the form of n=1 to ∞ (infinity).

## Q: How is an infinite series evaluated?

An infinite series can be evaluated by finding the sum of all the terms in the series. This can be done using various methods such as geometric series, telescoping series, and power series.

## Q: What is the difference between a convergent and a divergent infinite series?

A convergent infinite series is one in which the sum of all the terms approaches a finite value as the number of terms increases. In contrast, a divergent infinite series is one in which the sum of all the terms either approaches infinity or does not have a finite value.

## Q: What is the importance of evaluating an infinite series?

Evaluating an infinite series is important in many areas of mathematics, physics, and engineering. It allows us to find the sum of an infinite number of terms, which can be used to solve various real-world problems and make predictions.

## Q: How is an infinite series evaluated using a computer?

An infinite series can be evaluated using a computer by using algorithms and programming languages such as Python and MATLAB. These programs can handle large numbers and can calculate the sum of an infinite series with high accuracy.

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